Lie-Poisson groups: Remarks and examples (Q916801)

The authors discuss two possible definitions of complex structures on Lie-Poisson groups and give a complete classification of the isomorphism classes of complex Lie-Poisson structures on SL(2,\({\mathbb{C}})\). On the other hand, they give an algebraic characterization of a class of solutions of the Yang-Baxter equations, which contains the well-known Drinfeld solutions [\textit{V. G. Drinfel'd}, Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)]; in particular, they prove the existence of a nontrivial Lie-Poisson structure on any simply connected real semisimple Lie group G.